Problem

Complete the square for $R(p)=-p^{2}+18p$. Use the resulting vertex form to find the maximizing price, confirm that the price is admissible, and interpret both coordinates of the vertex with dollar units.

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Completing the square turns the revenue model’s optimum into the vertex of a downward-opening parabola. We’ll factor the negative leading coefficient, complete and compensate the square inside, distribute the sign carefully, then check the vertex input against the price domain and interpret input and output with their distinct dollar meanings.

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R(p) = −(p − 9)² + 81, so the maximum revenue is $81. The maximizing input price and its domain status are not given.

R(p) = −(p − 9)² + 81. The maximum occurs at admissible price p = $9 and is $81 of revenue, so the model predicts maximum revenue of $81 at a price of $9.

R(p) = −(p − 9)² + 81, so p = $9 gives $81 revenue. The price is not checked against the stated p ≥ 0 domain.

R(p) = −(p − 9)² + 81, so p = $9 of revenue and the maximum is 81 dollars of price. The input and output units are reversed.

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Complete the square for \(R(p)=-p^{2}+18p\). Use the resulting vertex form to find the maximizing price, confirm that the price is admissible, and interpret both coordinates of the vertex with dollar units.

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